Question

find the volume of the solid. for formulas containing π, give the exact answer and then an approximation using (\frac{22}{7}) for π.

the exact value of the volume is 12π cubic yards.
(simplify your answer. type an exact answer in terms of π.)

the approximate value of the volume is (square)
(simplify your answer. type a whole number.)

dropdown options: yards, square yards, cubic yards.
image of a cone with height 9 yd and radius 2 yd

Explanation:

Step1: Substitute π with 22/7

We know the exact volume is \( 12\pi \) cubic yards. Substitute \( \pi = \frac{22}{7} \), so the volume becomes \( 12\times\frac{22}{7} \).

Step2: Calculate the product

\( 12\times\frac{22}{7}=\frac{264}{7}\approx37.714 \), but we need a whole number approximation. Wait, maybe I made a mistake. Wait, the cone volume formula is \( V = \frac{1}{3}\pi r^2 h \). Wait, the radius is 2 yd, height is 9 yd. So exact volume: \( \frac{1}{3}\pi(2)^2(9)=\frac{1}{3}\pi\times4\times9 = 12\pi \), that's correct. Then approximate: \( 12\times\frac{22}{7}=\frac{264}{7}\approx37.71 \), but maybe the problem expects using the formula correctly. Wait, no, the exact is 12π, so approximate: 12(22/7) = 264/7 ≈ 37.71, but if we take whole number, maybe 38? Wait, no, 264 divided by 7 is 37.714..., so the approximate value (whole number) is 38? Wait, no, maybe I miscalculated. Wait, 1222=264, 264/7=37.714..., so the approximate volume (as a whole number) is 38? Wait, but let's check again. The formula for the volume of a cone is \( V=\frac{1}{3}\pi r^2 h \). Given r=2, h=9. So \( V=\frac{1}{3}\pi(2)^2(9)=\frac{1}{3}\pi\times4\times9 = 12\pi \). Then substituting \( \pi=\frac{22}{7} \), we get \( 12\times\frac{22}{7}=\frac{264}{7}\approx37.71 \), which rounds to 38? Wait, but maybe the problem expects 37 or 38. Wait, 22/7 is approximately 3.142857. 123.142857≈37.714, so the whole number approximation is 38? Wait, no, maybe the question has a typo, but according to the calculation, 12(22/7)=264/7≈37.71, so the approximate value (whole number) is 38? Wait, but let's do the calculation again. 2212=264, 264 divided by 7: 737=259, 264-259=5, so 37 + 5/7≈37.71, so the whole number is 38? Or maybe 37? Wait, the problem says "type a whole number", so we can round 37.71 to 38. Also, the unit is cubic yards, as volume is in cubic units.

Answer:

The approximate value of the volume is 38 cubic yards.